Problem
if i have $f:\mathbb{R}^2\rightarrow \mathbb{R}$ and $w:\mathbb{R}^3\rightarrow \mathbb{R}$ and for example if we have $f(x,y)=x^3+3xy^2$ $$ f(x,y)=x^3+3xy^2 $$ $$ x^3+3xy^2-f(x,y)=0 $$we can mark this as function of $w(x,y,f(x,y))$ $$ w(x,y,z)=x^3+3xy^2-f(x,y) $$ Is it true that:
$$ \nabla w(x,y,z)=\begin{bmatrix} \frac{\delta}{\delta x} x^3 \\ \frac{\delta}{\delta y} 3xy^2 \\ \frac{\delta}{\delta f(x,y)} (-f(x,y)) \end{bmatrix} $$ in every situation $\nabla(x,y,z)$ is normal vector for tangent plane of $f(x,y)$
This is something that my intuition would tell me this is correct but i don't have definitive proof of this. If someone could confirm this / confirm this is wrong that would be highly appreciated.